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ANALYTICAL MECHANICS
.ocity along any direction equals the rate at which distance .s covered along that direction. The velocity and its components evidently fulfill the
-elation_____
v=^vx*+vv*+v*.                        (Ill)
iVhen, as in the case of Fig. 48, the particle moves in the ;y-plane; z = 0, therefore
V=     V^/   +    Vy2.
Che direction of v, in this case, is given by
tan0= v,
(in')
(IV)
yhere 6 is the angle v makes with the ar-uxLs.
ILLUSTRATIVE EXAMPLR
Find the path, the velocity, and the components of the velocity of a •article which moves so that its position at any instant, is given by the blowing equations:
x = at,                                                        (a)
y = — Jf/r2.                                                (1>)
]liminating t between (a) and (b), we obtain
^     2(^
* ~   Ty/;
)r the equation of the path, therefore the path in a parabola, Fig. -I!). To find the component-velocities we differentiate (a) and (10 with aspect to the time.   This gives                                      V
x = a,                                   _^
y = —gt.
DISCUSSION. -— The horizontal cornpo-mt of the velocity is directed to the right id is constant, while the vertical corn-3nent is directed downwards and increases ; a constant rate.
We will see later that these equations present the motion of a body which is projected horizontally from an evated position.
I''KI. M.